a dipole subject to a flat metallic surfaceThe Green's function can be used to find the electric field or electrostatic potential for arbitrary charge distributions. You will learn that the polarization of a dipole in front of a flat metallic surface enormously affects the overall field.

Problem Statement

A dipole in front of a planar metallic surfaceUse the Green's function to calculate the electrostatic potential \(\phi\left(\mathbf{r}\right)\)  for a dipole \(\mathbf{p}\) at \(\mathbf{r}_{D}\) in front of a flat metallic surface terminated at \(z=0\). Determine  the main contributions to the potential for \(\left|\mathbf{r}_{D}\right|\ll\left|\mathbf{r}\right|\). Which has a stronger fall-of: \(x\)- or \(z\)-polarization? Can you explain this in simple words?


Background: Dipole Emitters in Front of Flat Metallic Surfaces

In this problem, we discuss the electrostatic dipole in the vicinity of a flat metal which is of huge interest both in applied and fundamental physics: think of a dipole antenna with a metal backplate or an emitting molecule. Of course such systems demand an understanding of full electrodynamics. The point is, however, that the main features will remain; \(x\)- and \(z\)-polarization will radiate differently.

It gets even more interesting if we do not have a perfect metal. Then, the electric field can penetrate the conductor and a closely placed dipole can excite surface waves as predicted by Sommerfeld more than one hundred years ago. Even though this effect was not the explanation for long-range radio transmissions, the theory of Sommerfeld is still at the core of plasmonics, the field that deals with these so-called surface plasmon polaritons. For much more information on dipoles in front of flat metallic surfaces please have a look at Lukas Novotny's freely available book chapter on the topic.

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